A:
Dear dear,
Other Miyazaki films aren't quite a kid-ish (and even his kid's films deal with mature themes like aging and man's relationship to nature).
Grave of the Fireflies is an earlier Miyazaki film that sledge hammers my brittle heart every time I see it.
Nausicaa of the Valley of the Wind has similar anti-war themes. It's set in the future when most of the earth is covered in acid forests. However, now that I think about it, if you didn't like
Princess Mononoke, you might not like
Nausicaa. If you want a cyberpunk police thriller, I highly recommend the Ghost in Shell movie
Solid State Society. It's not completely action - there is a complex plot - but the action it does have is high-quality. There's an earlier Ghost in Shell movie, created before the series (called
Ghost in Shell), which was named as an influence for
The Matrix, but I haven't seen it yet so I can't quite recommend it myself. I hear the Evangelion movies have some good action, and personally I liked the Escaflowne movie.
Here is a good article on some other anime movies.
There are four final fantasy movies.
Final Fantasy VII: Advent Children and
Final Fantasy: The Spirits Within are both computer animated. According to this
list of Final Fantasy Media, there are two other animated Final Fantasy films:
Final Fantasy: Legend of the Crystals and
Last Order: Final Fantasy VII.
-Whistler